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How to Prove that a Quadrilateral Is a Rhombus

You can use the following six methods to prove that a quadrilateral is a rhombus. The last three methods in this list require that you first show (or be given) that the quadrilateral in question is a parallelogram:

  • If all sides of a quadrilateral are congruent, then it’s a rhombus (reverse of the definition).

  • If the diagonals of a quadrilateral bisect all the angles, then it’s a rhombus (converse of a property).

  • If the diagonals of a quadrilateral are perpendicular bisectors of each other, then it’s a rhombus (converse of a property).

    Tip: To visualize this one, take two pens or pencils of different lengths and make them cross each other at right angles and at their midpoints. Their four ends must form a diamond shape — a rhombus.

  • If two consecutive sides of a parallelogram are congruent, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).

  • If either diagonal of a parallelogram bisects two angles, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).

  • If the diagonals of a parallelogram are perpendicular, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).

Here’s a rhombus proof for you. Try to come up with a game plan before reading the two-column proof.

image0.png image1.jpg

Statement 1:

image2.png

Reason for statement 1: Given.

Statement 2:

image3.png

Reason for statement 2: Opposite sides of a rectangle are congruent.

Statement 3:

image4.png

Reason for statement 3: Given.

Statement 4:

image5.png

Reason for statement 4: Like Divisions Theorem.

Statement 5:

image6.png

Reason for statement 5: All angles of a rectangle are right angles.

Statement 6:

image7.png

Reason for statement 6: All right angles are congruent.

Statement 7:

image8.png

Reason for statement 7: Given.

Statement 8:

image9.png

Reason for statement 8: A midpoint divides a segment into two congruent segments.

Statement 9:

image10.png

Reason for statement 9: SAS, or Side-Angle-Side (4, 6, 8)

Statement 10:

image11.png

Reason for statement 10: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Statement 11:

image12.png

Reason for statement 11: Given.

Statement 12:

image13.png

Reason for statement 12: If a triangle is isosceles, then its two legs are congruent.

Statement 13:

image14.png

Reason for statement 13: Transitivity (10 and 12).

Statement 14:

image15.png

Reason for statement 14: If a quadrilateral has four congruent sides, then it’s a rhombus.

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